Bounded Martian satellite relative motion

Abstract

Satellite relative motion around the Earth has been thoroughly studied during the last two decades. However, considerably less attention has been given to the study of satellite relative motion around Mars. As the cost of space technologies decreases and more space missions are within reach, formation flying missions around Mars have the potential to benefit future exploration missions launched to the Red Planet. A key parameter in such missions will be the frequency at which the spacecraft need to perform formation-keeping maneuvers to compensate for unwanted drifts due to differential perturbations. The Martian $J_3$ and $J_4$ gravitational harmonics are significant enough to warrant a dedicated investigation of bounded satellite relative motion configurations. In this study, we derive conditions for bounded satellite relative motion in non-critical inclinations around Mars, while considering its gravitational harmonics up to $J_4$. We first introduce a family of stable frozen orbits facilitating the implementation of formation flying and then apply differential nodal precession negation and differential periapsis rotation negation methods while considering the gravitational harmonics up to $J_4$. Using this procedure, we demonstrate how the secular growth of the relative distance can be arrested during long time intervals.

Appendix

$$\begin{aligned}&A_{\varOmega } = 84 (3 e_\mathrm{f}^{2} + 2) a^{2} \sin {i} J_{2} - 54 (15 \sin ^{2}{i} + 4) a e_\mathrm{f} r_\mathrm{eq} J_{3} \\&\qquad \quad + 165 (42 e_\mathrm{f}^{2} \sin ^{2}{i} - 23 e_\mathrm{f}^{2} + 7 \sin ^{2}{i} - 4) r_\mathrm{eq}^{2} \sin {i} J_{4}\\&E_{\varOmega } = 12 (3 e_\mathrm{f}^{2} - 8) a^{2} e_\mathrm{f} \sin {i} J_{2} + 6 (15 \sin ^{2}{i} - 4) a r_\mathrm{eq} J_{3} \\&\qquad \quad + (636 e_\mathrm{f}^{2} \sin ^{2}{i} - 345 e_\mathrm{f}^{2} - 16 \sin ^{2}{i} + 750) e_\mathrm{f} r_\mathrm{eq}^{2} \sin {i} J_{4} \\ {}&I_{\varOmega } = 12 (3 e_\mathrm{f}^{2} + 2) a^{2} \sin {i} J_{2} + 6 (15 - 30 \sin ^{2}{i} + 4 \csc ^{2}{i}) a e_\mathrm{f} r_\mathrm{eq} J_{3} \\ {}&\qquad \quad + 15 (126 e_\mathrm{f}^{2} \sin ^{3}{i} - 107 e_\mathrm{f}^{2} \sin {i} + 21 \sin ^{3}{i} - 18 \sin {i}) r_\mathrm{eq}^{2} J_{4} \\ {}&A_{\omega } = 84 (5 \sin ^{2}{i} - e_\mathrm{f}^{2} - 4) a^{2} e_\mathrm{f} J_{2} \\ {}&\qquad \quad + 54 (15 e_\mathrm{f}^{2} \sin {i}- 10 e_\mathrm{f}^{2} \sin ^{3}{i} - 4 e_\mathrm{f}^{2} \csc {i} - 5 \sin ^{3}{i} + 4 \sin {i}) a r_\mathrm{eq} J_{3} \\ {}&\qquad \quad + 165 (21 e_\mathrm{f}^{2} \sin ^{4}{i} - 34 \sin ^{2}{i} - 42 e_\mathrm{f}^{2} \sin ^{2}{i} + 19 e_\mathrm{f}^{2} + 28 \sin ^{4}{i} + 8) e_\mathrm{f} r_\mathrm{eq}^{2} J_{4} \\ {}&E_{\omega } = 12 (6 - 5 \sin ^{2}{i} - e_\mathrm{f}^{2}) a^{2} e_\mathrm{f}^{3} J_{2} \\ {}&\qquad \quad + 6 (20 e_\mathrm{f}^{2} \sin ^{3}{i} - 23 e_\mathrm{f}^{2} \sin {i} + 4 e_\mathrm{f}^{2} \csc {i} - 5 \sin ^{3}{i} + 4 \sin {i}) a r_\mathrm{eq} J_{3} \\ {}&\qquad \quad + 15 (21 e_\mathrm{f}^{2} \sin ^{4}{i} - 42 e_\mathrm{f}^{2} \sin ^{2}{i} + 19 e_\mathrm{f}^{2} - 70 \sin ^{4}{i} + 118 \sin ^{2}{i} - 46) e_\mathrm{f}^{3} r_\mathrm{eq}^{2} J_{4} \\ {}&I_{\omega } = 60 a^{2} e_\mathrm{f} \sin ^3{i} J_{2} \\ {}&\qquad \quad + 3 (4 e_\mathrm{f}^{2} + 15 e_\mathrm{f}^{2} \sin ^{2}{i} + 4 \sin ^{2}{i} - 30 e_\mathrm{f}^{2} \sin ^{4}{i} - 15 \sin ^{4}{i}) a r_\mathrm{eq} J_{3} \\ {}&\qquad \quad + 30 (28 \sin ^{3}{i} + 21 e_\mathrm{f}^{2} \sin ^{3}{i} - 21 e_\mathrm{f}^{2} \sin {i} - 17 \sin {i}) e_\mathrm{f} r_\mathrm{eq}^{2} \sin ^2{i} J_{4} \\ {}&A_M = -96 a^{4} e_\mathrm{f} + 168 (12 e_\mathrm{f}^{2} \sin ^{2}{i} - 8 e_\mathrm{f}^{2} + 3 \sin ^{2}{i} - 2) a^{2} e_\mathrm{f} r_\mathrm{eq}^{2} J_{2} \\ {}&\qquad \quad + 108 (36 e_\mathrm{f}^{2} - 45 e_\mathrm{f}^{2} \sin ^{2}{i} + 5 \sin ^{2}{i} - 4) a r_\mathrm{eq}^{3} \sin {i} J_{3} \\ {}&\qquad \quad + 165 (252 e_\mathrm{f}^{2} \sin ^{4}{i} - 276 e_\mathrm{f}^{2} \sin ^{2}{i} + 48 e_\mathrm{f}^{2} - 7 \sin ^{4}{i} + 6 \sin ^{2}{i}) e_\mathrm{f} r_\mathrm{eq}^{4} J_{4} \\ {}&E_M = 48 (2 - 3 \sin ^{2}{i}) a^{2} e_\mathrm{f}^{3} J_{2} + 3 (45 e_\mathrm{f}^{2} \sin ^{2}{i} - 36 e_\mathrm{f}^{2} + 5 \sin ^{2}{i} - 4) a r_\mathrm{eq} \sin {i} J_{3} \\ {}&\qquad \quad - 90 (21 \sin ^{4}{i} + 23 \sin ^{2}{i} - 4) e_\mathrm{f}^{3} r_\mathrm{eq}^{2} J_{4} \\ {}&I_M = -36 (4 e_\mathrm{f}^{2} + 1) a^{2} e_\mathrm{f} \sin {i} J_{2} + 3(135 e_\mathrm{f}^{2} \sin ^{2}{i} - 36 e_\mathrm{f}^{2} - 15 \sin ^{2}{i} + 4) a r_\mathrm{eq} J_{3} \\ {}&\qquad \quad + 15 (138 e_\mathrm{f}^{2} - 252 e_\mathrm{f}^{2} \sin ^{2}{i} + 7 \sin ^{2}{i} - 3) e_\mathrm{f} r_\mathrm{eq}^{2} \sin {i} J_{4}&\end{aligned}$$